Related papers: Counting arithmetic lattices and surfaces

We prove there are exactly 16 arithmetic lattices of hyperbolic 3-space which are generated by two elements of finite orders p and q with p,q at least six. We also verify a conjecture of H.M. Hilden, M.T. Lozano, and J.M. Montesinos…

The univariate Ehrhart and $h^*$-polynomials of lattice polytopes have been widely studied. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and $h^*$-polynomials of lattice polytropes, which…

Volume estimates of metric balls in manifolds find diverse applications in information and coding theory. In this paper, some new results for the volume of a metric ball in unitary group are derived via various tools from random matrix…

Let $\Gamma$ denote the modular group $SL(2,\Bbb Z)$ and $C_n(\Gamma)$ the number of congruence subgroups of $\Gamma$ of index at most $n$. We prove that $\lim\limits_{n\to \infty} \frac{\log C_n(\Gamma)}{(\log n)^2/\log\log n} =…

Let $G$ be a simply connected solvable Lie group with a lattice $\Gamma$ and $N$ the nilradical of $G$. For a complex valued representation $\rho: G\to GL(V_{\rho})$ such that the restriction $\rho_{|_{N}}$ is unipotent, as an advanced…

We estimate the upper bound for the $\ell^{\infty}$-norm of the volume form on $\mathbb{H}^2\times\mathbb{H}^2\times\mathbb{H}^2$ seen as a class in…

Let $\Gamma$ be a finite graph and let $A(\Gamma)$ be the corresponding right-angled Artin group. We characterize the Hamiltonicity of $\Gamma$ via the structure of the cohomology algebra of $A(\Gamma)$. In doing so, we define and develop a…

Let $\Gamma_1$ and $\Gamma_2$ be two lattices of finite covolume in a semisimple Lie group $G$. We prove a spectral rigidity result for the representation spectra of the right regular representations $L^2(\Gamma_1 \backslash G)$ and…

This paper contains a purely topological theorem and a geometric application. The topological theorem states that if M is a simple closed orientable 3-manifold such that \pi_1(M) contains a genus g surface group and H_1(M;Z/2Z) has rank at…

Hyperideal tetrahedra are the fundamental building blocks of hyperbolic 3-manifolds with geodesic boundary. The study of their geometric properties (in particular, of their volume) has applications also in other areas of low-dimensional…

We consider closed hypersurfaces smoothly immersed in hyperbolic manifolds up to homotopy and commensurability. We prove that if a closed hyperbolic manifold $M$ contains a sequence of asymptotically geodesic hypersurfaces, then $\pi_1(M)$…

We develop the theory of $L^2$-torsion of an automorphism of a group and compute it for every automorphism of a group which is hyperbolic and one-ended relative to a finite collection of virtually polycyclic groups. We also prove a…

Any action of a group $\Gamma$ on $\mathbb H^3$ by isometries yields a class in degree three bounded cohomology by pulling back the volume cocycle to $\Gamma$. We prove that the bounded cohomology of finitely generated Kleinian groups…

Consider the configuration spaces of manifolds. We give a precise formula for the integral cohomological dimension (the degree of top non-trivial integral cohomology group) of unordered configuration spaces of manifolds with non-trivial…

If $\Gamma<\mathrm{PSL}(2,\mathbb{C})$ is a lattice, we define an invariant of a representation $\Gamma\rightarrow \mathrm{PSL}(n,\mathbb{C})$ using the Borel class $\beta(n)\in…

In this paper we show how to use elementary methods to prove that the volume of Sl_k R / Sl_k Z is zeta(2) * zeta(3) * ... * zeta(k) / k. Using a version of reduction theory presented in this paper, we can compute the volumes of certain…

There is a natural left and right invariant Haar measure associated with the matrix groups GL${}_N(\mathbb R)$ and SL${}_N(\mathbb R)$ due to Siegel. For the associated volume to be finite it is necessary to truncate the groups by imposing…

Given the fundamental group $\Gamma$ of a finite-volume complete hyperbolic $3$-manifold $M$, it is possible to associate to any representation $\rho:\Gamma \rightarrow \text{Isom}(\mathbb{H}^3)$ a numerical invariant called volume. This…

Let $X=S\times E \times B$ be the metric product of a symmetric space $S$ of noncompact type, a Euclidean space $E$ and a product $B$ of Euclidean buildings. Let $\Gamma$ be a discrete group acting isometrically and cocompactly on $X$. We…

We study the asymptotic behaviour of the cohomology of subgroups $\Gamma$ of an algebraic group $G$ with coefficients in the various irreducible rational representations of $G$ and raise a conjecture about it. Namely, we expect that the…